Let A 1 3 1 5 Find an invertible matrix P and a diagonal ma

Let A = [-1 3 -1 -5]. Find an invertible matrix P and a diagonal matrix D such that A = PDP^-1.

Solution

The characteristic equation of A is det (A- I2) = 0 or, 2+6 +8= 0 (+4)(+2) = 0. Thus, the eigenvalues of A are 1 =-4 and 2 = -2. Further, the eigenvector of A corresponding to the eigenvalue is solution to the equation (A- I2)X = 0. Thus, the eigenvector of A corresponding to the eigenvalue -4 is solution to the equation (A+4I2)X = 0. We will reduce A+4I2 to its RREF as under:

Multiply the 1st row by 1/3

Add 1 times the 1st row to the 2nd row

Then the RREf of A+4I2 is

1

1

0

0

If X = (x,y)T, then the equation (A+4I2)X = 0 is equivalent to x+y = 0 so that X = (-y,y)T = y(-1,1)T. Hence the eigenvector of A corresponding to the eigenvalue -4 is (-1,1)T. Similarly, the eigenvector of A corresponding to the eigenvalue -2 is (-3,1)T. Then D =

-4

0

0

-2

and P =

-1

-3

1

1

1	1
0	0